8 LEBESGUE'S CONVERGENCE THEOREMS 127
The usefulness of this theorem is that it does not require the convergence
to be uniform (not even on compact subsets). In particular:
(13.8.5) Let (/„) be a sequence of integrable functions^ such that the series with
general term \fn\ djji is convergent. Then the series with general term fn(x) is
almost everywhere absolutely convergent in R, and if weputf(x) = £ fn(x\
the function f (defined almost everywhere) is integrable and we have
(22.214.171.124) j/(x)^W = n|;J/B(x
(" term-by-term integration of a series of integrable functions").
For each integer N ^ 0, let #N = £ \fn\. The sequence (#N) is an increasing
sequence of integrable functions, and
((13.7.4) and (13.7.3)). By (13.8.1), the function # = lim #N = £ |/J is
integrable. This implies first of all that g is finite almost everywhere (13.6.4),
in other words that the series with general term \fn(x)\ is convergent almost
everywhere, which proves the first assertion ((5.3.1) and (5.3.2)). Moreover,
for all N we have ]T \fn\ ^ g, from which by (13.8.4) follow the integrability
of /and the relation (126.96.36.199).
(13.8.6) (i) Let E be a metric space, (x, z)\-^f(x, z) a mapping ofX x E into
K, and z0 a point ofE. Suppose that
(1) for each z e E, the function x\-+f(x, z) is integrable ;
(2) for almost all x e X, the function z\-^>f(x, z) is continuous at ZQ ;
(3) there exists an integrable function g ^ 0 such that., for all z e E, we
have \f(x, z)\ ^ g(x) almost everywhere in X.
Then h(z) = \f(x, z) dfi(x) is continuous at ZQ .
(ii) Suppose furthermore that E is an open interval in R and that f satisfies
the following conditions:
(4) for almost all xeX, the function ZH->/(X, z) is finite and admits a
derivative D2/(x, z);re, for alln. Then f is integrable and